Algebraic Number Theory: A Brief Introduction

✍ Scribed by J.S. Chahal

• Publisher: Chapman and Hall/CRC
• Year: 2021
• Tongue: English
• Leaves: 167
• Series: Textbooks in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

This book offers the basics of algebraic number theory for students and others who need an introduction and do not have the time to wade through the voluminous textbooks available. It is suitable for an independent study or as a textbook for a first course on the topic.

The author presents the topic here by first offering a brief introduction to number theory and a review of the prerequisite material, then presents the basic theory of algebraic numbers. The treatment of the subject is classical but the newer approach discussed at the end provides a broader theory to include the arithmetic of algebraic curves over finite fields, and even suggests a theory for studying higher dimensional varieties over finite fields. It leads naturally to the Weil conjecture and some delicate questions in algebraic geometry.

About the Author

Dr. J. S. Chahal is a professor of mathematics at Brigham Young University. He received his Ph.D. from Johns Hopkins University and after spending a couple of years at the University of Wisconsin as a post doc, he joined Brigham Young University as an assistant professor and has been there ever since. He specializes and has published several papers in number theory. For hobbies, he likes to travel and hike. His book, Fundamentals of Linear Algebra, is also published by CRC Press.

✦ Table of Contents

Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Preface
1. Genesis: What Is Number Theory?
1.1. What Is Number Theory?
1.2. Methods of Proving Theorems in Number Theory
2. Review of the Prerequisite Material
2.1. Basic Concepts
2.2. Galois Extensions
2.3. Integral Domains
2.4. Factoring Rational Primes in Z[i]
3. Basic Concepts
3.1. Generalities
3.2. Algebraic Integers
3.3. Integral Bases
3.4. Quadratic Fields
3.5. Unique Factorization Property for Ideals
3.6. Ideal Class Group and Class Number
4. Arithmetic in Relative Extensions
4.1. Criterion for Rami cation
4.2. Review of Commutative Algebra
4.3. Relative Discriminant for Rings
4.4. Direct Product of Rings
4.5. Nilradical
4.6. Reduced Rings
4.7. Discriminant and Rami cation
5. Geometry of Numbers
5.1. Lattices in Rn
5.2. Minkowski's Lemma on Convex Bodies
5.3. Logarithmic Embedding
5.4. Units of a Quadratic Field
5.5. Estimates on the Discriminant
6. Analytic Methods
6.1. Preliminaries
6.2. The Regulator of a Number Field
6.3. Fundamental Domains
6.4. Zeta Functions
6.4.1. The Riemann Zeta Function
6.4.2. A Partial Zeta Function
6.4.3. The Dedekind Zeta Function
7. Arithmetic in Galois Extensions
7.1. Hilbert Theory
7.2. Higher Rami cation Groups
7.3. The Frobenius Map
7.4. Rami cation in Cyclic Extensions
7.5. The Artin Symbol
7.6. Quadratic Fields
7.7. The Artin Map
8. Cyclotomic Fields
8.1. Cyclotomic Fields
8.2. Arithmetic in Cyclotomic Fields
9. The Kronecker-Weber Theorem
9.1. Gauss Sums
9.2. Proof of the Kronecker-Weber Theorem
10. Passage to Algebraic Geometry
10.1. Valuations
10.2. Zeta Functions of Curves over Finite Fields
10.3. Riemann Hypothesis for Elliptic Curves over Finite
11. Epilogue: Fermat's Last Theorem
11.1. Fermat's Last Theorem
11.2. An Alternative Approach to Proving FLT
Bibliography
Index